## Transcribed Text

Question 6 2011
(a) Prove that every continuous function f : [0, 1] → [0, 1] has a fixed point, that is, the
equation f(x) = x has at least one solution x ∈ [0, 1].
(b) Give an example of a continuous function f :]0, 1] →]0, 1] which does not have a fixed
point.
(c) Prove that no continuous surjective function f :]0, 1] → R can be injective.
Question 4 2009
(a) Show that the topological space (X, T) is Hausdorff (T2) if and only if the diagonal
(x, x)|x ∈ X is closed in X × X with the product topology.
(b) Let (X, T) and (Y, U) be topological spaces. Prove that f : X → Y is continuous if
and only if for each A ⊆ X, f(A¯) ⊆ f(A).
Question 7 2010
(a) Prove that a compact Hausdorff space is normal.
(b) Prove that there is no continuous bijection f : [0, 1] → [0, 1[.
Question 6 2010
Decide which of the following statements are true and which are false. Provide a proof
for those which are true. Give a counter-example for those which are false.
- A closed subset of a compact space is compact.
- Every contracting map f :]0, 1] →]0, 1] has a unique fixed point, where 0, 1] is taken
with its Euclidean metric.
Question 1 2010
Let X be the set of all absolutely convergent series of real numbers, so that
X = {(xn)n∈N|xn ∈ R for all n ∈ N and X
n∈N
|xn| converges}.
Decide whether
ρ : X × X → R
+
0
((xn)n∈N,(yn)n∈N) 7−→ sup
n∈N
|xn − yn|
determines a metric on X. Carefully justify your answer.

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